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Circular Measurement of Angle

Prove that ...

Question

Prove that $cot \frac{π}{20} . cot \frac{3 π}{20} . cot \frac{5 π}{20} . cot \frac{7 π}{20} . cot \frac{9 π}{20} = 1$

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Solution

L.H.S $= cot \frac{π}{20} . cot \frac{3 π}{20} . cot \frac{5 π}{20} . cot \frac{7 π}{20} . cot \frac{9 π}{20}$

$= cot 9^{\circ} . cot 27^{\circ} . cot 45^{\circ} . cot 63^{\circ} . cot 81^{\circ}$

$= cot 9^{\circ} . cot 27^{\circ} .1 . cot (90^{\circ} - 27^{\circ}) . cot (90^{\circ} - 9^{\circ})$

$= cot 9^{\circ} . cot 27^{\circ} . tan 27^{\circ} . tan 9^{\circ}$

$= (cot 9^{\circ} tan 9^{\circ}) (cot 27^{\circ} tan 27^{\circ})$

$= 1 \times 1 = 1$

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